I keep it straight in my mind by thinking of this: free cash flow to the firm is the cash flow available to ALL providers of capital, let it be creditors, shareholders or preferred shareholders. Investment into working capital and investment into fixed capital are the cash investments we make for our operational assets to keep on going. You make investments in those before you pay off creditors or shareholders, in a perfect world.
A decrease in fixed capital could lead to a negative figure in the equation, adding to FCFF if you sold the assets. That frees up cash available to all creditors and shareholders. A decrease in FC due to depreciation will not since, as you know, depreciation is a non-cash charge. To circle it all up, Ending FC = Beg FC + FC Investment - Depreciation. What they are asking for in the FCFF equation in the FC Investment (use of cash, which is why it’s subtracted in the FCFF equation), not Ending FC - Beg FC, although in a theoretically perfect world with no depreciation, they could be the same thing.
The same logic goes with working capital.